You will probably find the former British Islands use the old British legislation which is very strict. It wasn't until the UK became part of the EU some of the more primitive laws were relaxed. Dogs now have to carry a pet passport which contains their current vacination status and microchip data. See information on the PETS (Pet travel scheme) at
Defra, UK - Animal health and welfare - Pet Travel Scheme - Dogs, cats and ferrets this will show you the territories and caribbean islands that operate this scheme.
It's a nightmare getting pets back in to the UK, you can only arrive on certain transport (ferries, aeroplanes), at specified ports, and vacinations must have been carried out in the last 48 hours prior to arrival. As an example if I want to sail back to the UK with the dog, I have to drop off the wife and dog at a French ferry port, get a vet to vacinate the dog and stamp its passport, they then have to get on a ferry and I have to single hand cross the channel to meet them at the other end, once cleared they can then get back on our boat. If all of this isn't met the dog is put into quarantine for 6 months.
A royal pain in the backside.